Freudenthal-Kantor triple system
A triple system considered for constructing all simple Lie algebras (cf. Lie algebra), and introduced as an algebraic system which is a generalization both of the algebraic systems appearing in the metasymplectic geometry developed by H. Freudenthal and of a generalized Jordan triple system of second order developed by I.L. Kantor.
Recall that a triple system is a vector space over a field \Phi together with a \Phi-trilinear mapping V \times V \times V \rightarrow V.
For \epsilon = \pm 1, a vector space U ( \varepsilon ) over a field \Phi with the trilinear product \langle x y z \rangle is called a Freudenthal–Kantor triple system if
\begin{equation} \tag{a1} \langle a b \langle c d e \rangle \rangle = \langle \langle a b c \rangle \rangle + \varepsilon \langle c \langle b a d \rangle e \rangle + \langle c d \langle a b e \rangle \rangle, \end{equation}
\begin{equation} \tag{a2} K ( L ( a , b ) c , d ) + K ( c , L ( a , b ) d ) + K ( a , K ( c , d ) b ) = 0, \end{equation}
where L ( a , b ) c = \langle a b c \rangle and K ( a , b ) c = \langle a c b \rangle - \langle b c a \rangle.
In particular, a Freudenthal–Kantor triple system U ( \varepsilon ) is said to be balanced if there exists a bilinear form \langle \, .\, ,\, . \, \rangle such that K ( a , b ) = \langle a , b \rangle \operatorname{Id}, for all a,b \in U ( \varepsilon ).
This balancing property is closely related to metasymplectic geometry.
Note that if \varepsilon = - 1 and K ( a , b ) \equiv 0 (identically), then the Freudenthal–Kantor triple system reduces to a Jordan triple system.
As the notion of a Freudenthal–Kantor triple system includes the notions of a generalized Jordan triple system of second order, a structurable algebra, and an Allison–Hein triple system, it is useful in obtaining all Lie algebras, without the use of root systems and Cartan matrices.
Let V be a vector space with a bilinear form \langle x , y \rangle = - \varepsilon \langle y , x \rangle. Then V is a Freudenthal–Kantor triple system with respect to the triple product \langle x y z \rangle : = \langle y , z \rangle x. In particular, it is important that the linear span \mathbf{k} : = \{ K ( a , b ) \} _ { \text{span} } of the set K ( a , b ) makes a Jordan triple system of ( \text { End } U ( \varepsilon ) ) ^ { + } with respect to the triple product \{ A B C \} : = 1 / 2 ( A B C + C B A ).
Let U ( \varepsilon ) be a Freudenthal–Kantor triple system. The vector space U ( \varepsilon ) \oplus U ( \varepsilon ) becomes a Lie triple system with respect to the triple product defined by
\begin{equation*} \left[ \left( \begin{array} { l } { a } \\ { b } \end{array} \right) \left( \begin{array} { l } { c } \\ { d } \end{array} \right) \left( \begin{array} { l } { e } \\ { f } \end{array} \right) \right] : = \end{equation*}
\begin{equation*} := \left( \begin{array} { c c } { L ( a , d ) - L ( c , b ) } & { K ( a , c ) } \\ { - \varepsilon K ( b , d ) } & { \varepsilon ( L ( d , a ) - L ( b , c ) ) } \end{array} \right) \left( \begin{array} { l } { e } \\ { f } \end{array} \right). \end{equation*}
Using this, one can obtain the Lie triple system U ( \varepsilon ) \oplus U ( \varepsilon ) associated with U ( \varepsilon ); it is denoted be T ( \varepsilon ).
Using the concept of the standard embedding Lie algebra L ( \varepsilon ) = \operatorname { Inn } \operatorname { Der } T ( \varepsilon ) \oplus T ( \varepsilon ) associated with a Lie triple system T ( \varepsilon ), one can obtain the construction of L ( \varepsilon ) associated with a Freudenthal–Kantor triple system U ( \varepsilon ). In fact, put
L_{2} equal to the linear span of the endomorphisms
\begin{equation*} \left( \begin{array} { c c } { 0 } & { K ( a , b ) } \\ { 0 } & { 0 } \end{array} \right); \end{equation*}
L _ { 1 } : = U ( \varepsilon ) \oplus ( 0 );
L_{ - 1} : = ( 0 ) \oplus U ( \varepsilon );
L_0 equal to the linear span of the endomorphisms
\begin{equation*} \left( \begin{array} { c c } { L ( a , b ) } & { 0 } \\ { 0 } & { \varepsilon L ( b , a ) } \end{array} \right); \end{equation*}
L_{ - 2} equal to the linear span of the endomorphisms
\begin{equation*} \left( \begin{array} { r r } { 0 } & { 0 } \\ { - \varepsilon K ( c , d ) } & { 0 } \end{array} \right). \end{equation*}
Then one obtains the decomposition
\begin{equation*} L ( \varepsilon ) = L _ { - 2 } \bigoplus L _ { - 1 } \bigoplus L _ { 0 } \bigoplus L _ { 1 } \bigoplus L _ { 2 }, \end{equation*}
and, more precisely,
\begin{equation*} \left[ \left( \begin{array} { c c } { \text{Id} } & { 0 } \\ { 0 } & { - \text{Id} } \end{array} \right) , L _ { i } \right] = i L _ { i } ( - 2 \leq i \leq 2 ). \end{equation*}
These results imply the dimensional formula
\begin{equation*} \operatorname { dim } _ { \Phi } L ( \varepsilon ) = 2 ( \operatorname { dim } _ { \Phi } U ( \varepsilon ) + \operatorname { dim } _ { \Phi } \{ K ( x , y ) \} _ { \operatorname { span } } )+ \end{equation*}
\begin{equation*} + \operatorname { dim } _ { \Phi } \{ L ( x , y ) \} _ { \operatorname { span } } = \end{equation*}
\begin{equation*} = \operatorname { dim } _ { \Phi } T ( \varepsilon ) + \operatorname { dim } _ { \Phi } \operatorname { Inn } \operatorname { Der } T ( \varepsilon ). \end{equation*}
This algebra L ( \varepsilon ) is called the Lie algebra associated with U ( \varepsilon ).
The concepts of a triple system and a supertriple system are important in the theory of quarks and Yang–Baxter equations.
Note that a "triple system" in the sense discussed above is totally different from "triple system" in combinatorics (see, e.g., Steiner triple system).
References
[a1] | H. Freudenthal, "Beziehungen der E _ { 7 } und E _ { 8 } zur Oktavenebene I–II" Indag. Math. , 16 (1954) pp. 218–230; 363–386 |
[a2] | N. Kamiya, "The construction of all simple Lie algebras over C from balanced Freudenthal–Kantor triple systems" , Contributions to General Algebra , 7 , Hölder–Pichler–Tempsky, Wien (1991) pp. 205–213 |
[a3] | N. Kamiya, "On Freudenthal–Kantor triple systems and generalized structurable algebras" , Non-Associative Algebra and Its Applications , Kluwer Acad. Publ. (1994) pp. 198–203 |
[a4] | N. Kamiya, S. Okubo, "On \delta-Lie supertriple systems associated with ( \varepsilon , \delta )-Freudenthal–Kantor supertriple systems" Proc. Edinburgh Math. Soc. , 43 (2000) pp. 243–260 |
[a5] | I.L. Kantor, "Models of exceptional Lie algebras" Soviet Math. Dokl. , 14 (1973) pp. 254–258 |
[a6] | S. Okubo, "Introduction to octonion and other non-associative algebras in physics" , Cambridge Univ. Press (1995) |
[a7] | K. Yamaguti, "On the metasymplectic geometry and triple systems" Surikaisekikenkyusho Kokyuroku, Res. Inst. Math. Sci. Kyoto Univ. , 306 (1977) pp. 55–92 (In Japanese) |
Freudenthal-Kantor triple system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Freudenthal-Kantor_triple_system&oldid=51660